Biometry for Forestry and Environmental Data-With Examples in R

425 Pages·2020·10.136 MB·\425

by  Lauri Mehtatalo (Author), Juha Lappi (Author)

Biometry for Forestry and
Environmental Data
Chapman&Hall/CRC
AppliedEnvironmentalSeries
Series Editors
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Alexandra Schmidt, Universidade Federal do Rio de Janero
Richard L. Smith, University of North Carolina
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Biometry for Forestry and Environmental Data with Examples in R
Lauri Mehtätalo, Juha Lappi
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Biometry for Forestry and
Environmental Data
with Examples in R
Lauri Mehtätalo
Juha Lappi
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Contents
Preface xi
1 Introduction 1
2 Random Variables 5
2.1 Introduction to random variables . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Sources of randomness . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Univariate and multivariate random variables . . . . . . . . . . . . . 6
2.2 Univariate random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Sample space and support set . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Distribution function and density . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Transformations of random variables . . . . . . . . . . . . . . . . . . 14
2.2.4 Expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.5 Variance and standard deviation . . . . . . . . . . . . . . . . . . . . 16
2.2.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Multivariate random variables . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Joint, conditional and marginal distributions . . . . . . . . . . . . . 18
2.3.2 Bivariate transformations . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 Independence and dependence . . . . . . . . . . . . . . . . . . . . . 21
2.3.4 Covariance and correlation . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Calculation rules for expected value, variance and covariance . . . . . . . . 23
2.5 Conditional and marginal expected value and variance . . . . . . . . . . . . 24
2.6 Multivariate normal distribution . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Quantile function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Weighted and truncated distributions . . . . . . . . . . . . . . . . . . . . . 34
2.9 Compound distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.10 Sums of independent random variables . . . . . . . . . . . . . . . . . . . . 36
2.11 Transformations of a standard normal variable . . . . . . . . . . . . . . . . 38
2.11.1 χ2-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.11.2 Student’s t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.11.3 F-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.12 Random functions: stochastic and spatial processes . . . . . . . . . . . . . 41
3 Statistical Modeling, Estimation and Prediction 43
3.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 The role of assumptions in modeling . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Scales of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Estimation using observed data . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 Comparing alternative estimators . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Estimating expected value of any distribution . . . . . . . . . . . . . 47
Box: Least Squares estimation of µ . . . . . . . . . . . . . . . . . . . 48
3.4.3 Estimating variance and covariance of any distribution . . . . . . . . 50
v
vi Contents
3.4.4 Estimating any parameter of a specific distribution . . . . . . . . . . 51
3.4.5 Bayesian estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.1 Comparing alternative predictors . . . . . . . . . . . . . . . . . . . . 59
3.5.2 Linear prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.6 Model comparison and confidence intervals . . . . . . . . . . . . . . . . . . 62
3.6.1 Testing of hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.2 Multiple tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Box: Publication bias . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.3 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.4 Information theoretic model comparison . . . . . . . . . . . . . . . . 65
4 Linear Model 67
4.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Model for an individual observation . . . . . . . . . . . . . . . . . . 67
4.1.2 The model for all data . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 The systematic part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 Identifiability and multicollinearity . . . . . . . . . . . . . . . . . . . 72
4.2.2 Formulating the systematic part . . . . . . . . . . . . . . . . . . . . 72
4.2.3 Interpretation of regression coefficients . . . . . . . . . . . . . . . . . 74
4.2.4 Categorical predictors . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 The random part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Modeling inconstant error variance . . . . . . . . . . . . . . . . . . . 83
4.3.2 Modeling dependence using an autocorrelation function . . . . . . . 83
4.3.3 Variance-covariance structures in the model for all data . . . . . . . 84
4.3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Ordinary least squares (OLS) . . . . . . . . . . . . . . . . . . . . . . 89
Box: Stein’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Box: Ridge regression and LASSO . . . . . . . . . . . . . . . . . . . 91
4.4.2 Generalized least squares (GLS) . . . . . . . . . . . . . . . . . . . . 92
4.4.3 Maximum likelihood (ML). . . . . . . . . . . . . . . . . . . . . . . . 94
Box: BLUE or minimum RMSE? . . . . . . . . . . . . . . . . . . . . 95
4.4.4 Restricted maximum likelihood (REML) . . . . . . . . . . . . . . . . 95
Box: Warnings about model comparison using REML . . . . . . . . 97
4.5 Does the model fit? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.1 Model residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.2 Graphical evaluation of the fit . . . . . . . . . . . . . . . . . . . . . 100
4.5.3 Numerical criteria for model fit . . . . . . . . . . . . . . . . . . . . . 111
4.6 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6.1 Hypothesis tests of nested models . . . . . . . . . . . . . . . . . . . 113
4.6.2 Wald’s F- and χ2 tests of β . . . . . . . . . . . . . . . . . . . . . . . 114
4.6.3 Normality of residual errors in testing . . . . . . . . . . . . . . . . . 115
4.6.4 Marginal and sequential testing procedures . . . . . . . . . . . . . . 116
4.6.5 The effect of estimation error in V . . . . . . . . . . . . . . . . . . . 117
4.6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6.7 t-tests of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.6.8 Likelihood ratio - a general large-sample test procedure for nested
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.9 Comparing non-nested models . . . . . . . . . . . . . . . . . . . . . 125
Contents vii
4.6.10 Confidence intervals for model parameters . . . . . . . . . . . . . . . 125
4.7 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.7.1 Prediction of an uncorrelated observation . . . . . . . . . . . . . . . 126
4.7.2 Prediction of a correlated observation . . . . . . . . . . . . . . . . . 129
5 Linear Mixed-Effects Models 131
Box: Pseudo-replication . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.1.1 The variance component model . . . . . . . . . . . . . . . . . . . . . 133
5.1.2 Mixed-effects model with a random intercept . . . . . . . . . . . . . 136
5.1.3 Multiple random effects . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1.4 Random effects for categorical predictors . . . . . . . . . . . . . . . 142
5.2 Matrix formulation of the model . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2.1 Matrix formulation for single group. . . . . . . . . . . . . . . . . . . 145
5.2.2 Relaxing the assumptions on residual variance . . . . . . . . . . . . 147
5.2.3 Matrix formulation for all data . . . . . . . . . . . . . . . . . . . . . 147
5.3 Estimation of model parameters . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.1 Estimating variance components with ANOVA methods . . . . . . . 151
5.3.2 Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Box: Negative within-group correlation. . . . . . . . . . . . . . . . . 153
Box: Swamy’s GLS estimator . . . . . . . . . . . . . . . . . . . . . . 154
5.3.3 Restricted maximum likelihood . . . . . . . . . . . . . . . . . . . . . 155
5.3.4 More about the estimation of R . . . . . . . . . . . . . . . . . . . . 156
5.3.5 Bayesian estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4 Prediction at group level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.4.1 Prediction of random effects . . . . . . . . . . . . . . . . . . . . . . . 158
Box: Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.4.2 Group-level prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Box: Henderson’s mixed model equations . . . . . . . . . . . . . . . 162
5.4.3 A closer look at the variance component model . . . . . . . . . . . . 165
5.4.4 Fixed or random group effect? . . . . . . . . . . . . . . . . . . . . . 166
5.5 Evaluation of model fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Box: Plotting standard errors . . . . . . . . . . . . . . . . . . . . . . 169
5.5.1 The systematic part . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.5.2 The random part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.5.3 Coefficient of determination for mixed-effect models . . . . . . . . . 174
5.6 Tests and confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.6.1 Hypothesis tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.6.2 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.7 Reporting a fitted mixed-effects model . . . . . . . . . . . . . . . . . . . . . 182
6 More about Linear Mixed-Effects Models 183
6.1 Nested grouping structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.1.1 A nested two-level model . . . . . . . . . . . . . . . . . . . . . . . . 183
6.1.2 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.1.3 More than two nested levels . . . . . . . . . . . . . . . . . . . . . . . 189
6.2 Crossed grouping structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.3 Between-group and within-group effects . . . . . . . . . . . . . . . . . . . . 198
6.4 Population-averaged prediction . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.5 Prediction of correlated observation using a mixed model . . . . . . . . . . 205
viii Contents
6.6 Applying the mixed model BLUP in a new group without re-estimating the
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7 Nonlinear (Mixed-Effects) Models 209
7.1 Nonlinear model and nonlinear relationship . . . . . . . . . . . . . . . . . . 209
7.2 Nonlinear fixed-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Box: Taylor approximation . . . . . . . . . . . . . . . . . . . . . . . 211
7.2.2 Explaining variability in φ through second-order predictors . . . . . 214
Box: Finney’s second-level parameters . . . . . . . . . . . . . . . . . 216
7.2.3 Parameter estimation when var(e)=σ2I . . . . . . . . . . . . . . . 216
7.2.4 Initial guesses of parameters. . . . . . . . . . . . . . . . . . . . . . . 218
7.2.5 Inference and model diagnostics. . . . . . . . . . . . . . . . . . . . . 218
7.2.6 Parameter estimation and inference when var(e)=σ2V . . . . . . . 221
7.3 Nonlinear mixed-effects models . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.3.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.3.2 Parameter estimation and inference . . . . . . . . . . . . . . . . . . 225
7.3.3 Convergence problems . . . . . . . . . . . . . . . . . . . . . . . . . . 233
7.3.4 Multiple levels of grouping . . . . . . . . . . . . . . . . . . . . . . . 234
7.3.5 Prediction of random effects and y-values . . . . . . . . . . . . . . . 235
7.3.6 Principal components of random parameters . . . . . . . . . . . . . 241
8 Generalized Linear (Mixed-Effects) Models 245
Box: Transformations or GLMM? . . . . . . . . . . . . . . . . . . . . 246
8.1 Common parent distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.1.1 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.1.2 Bernoulli and binomial distribution . . . . . . . . . . . . . . . . . . . 247
8.1.3 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.1.4 Link and mean functions . . . . . . . . . . . . . . . . . . . . . . . . 248
8.2 Generalized linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2.1 Formulation of the LM as a GLM . . . . . . . . . . . . . . . . . . . 249
8.2.2 Models for binary data. . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.2.3 Models for count data . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.2.4 GLM for the exponential family. . . . . . . . . . . . . . . . . . . . . 258
8.2.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.2.6 Evaluating the fit of a GLM . . . . . . . . . . . . . . . . . . . . . . . 262
8.2.7 Over- and underdispersion . . . . . . . . . . . . . . . . . . . . . . . . 265
8.2.8 Inference on GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.2.9 Zero-inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8.2.10 Exposure variables in count data . . . . . . . . . . . . . . . . . . . . 275
8.3 Generalized linear mixed-effects models . . . . . . . . . . . . . . . . . . . . 275
8.3.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
8.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.3.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
9 Multivariate (Mixed-Effects) Models 287
9.1 Why model systems? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
9.2 Seemingly unrelated regression models . . . . . . . . . . . . . . . . . . . . . 289
9.2.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.2.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.2.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Contents ix
9.2.4 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
9.3 Simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
9.4 Multivariate mixed-effects models . . . . . . . . . . . . . . . . . . . . . . . 300
9.4.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.4.2 Estimation and inference . . . . . . . . . . . . . . . . . . . . . . . . 302
9.4.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
9.5 Multivariate nonlinear and generalized linear (mixed) models . . . . . . . . 308
10 Additional Topics on Regression 309
10.1 Random regressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.1.1 Connection between OLS linear regression and BLP . . . . . . . . . 309
Box: Regression through the origin . . . . . . . . . . . . . . . . . . . 311
10.1.2 Random regressors in a linear model . . . . . . . . . . . . . . . . . . 312
10.2 Modeling nonlinear responses using transformations in y . . . . . . . . . . 314
10.2.1 Logarithmic regression . . . . . . . . . . . . . . . . . . . . . . . . . . 314
10.2.2 Correcting back transformation bias using the Taylor series . . . . . 316
10.2.3 Correcting back transformation bias using the two-point distribution 318
10.3 Predicting the mean of a nonlinear function over the distribution of x . . . 319
10.4 Modeling nonlinear responses using transformations in x . . . . . . . . . . 321
10.4.1 Regression splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Box: Interpolating splines . . . . . . . . . . . . . . . . . . . . . . . . 324
10.4.2 Smoothing splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
10.4.3 Multivariate splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
10.4.4 Second-order response surface . . . . . . . . . . . . . . . . . . . . . . 325
10.5 Generalized additive models . . . . . . . . . . . . . . . . . . . . . . . . . . 329
10.6 Modeling continuous proportions . . . . . . . . . . . . . . . . . . . . . . . . 330
10.7 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11 Modeling Tree Size 333
Box: Kernel smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . 334
11.1 A model for tree size in a single forest stand . . . . . . . . . . . . . . . . . 336
11.2 Fitting an assumed model to tree diameter data . . . . . . . . . . . . . . . 337
11.2.1 Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
11.2.2 The method of moments . . . . . . . . . . . . . . . . . . . . . . . . . 339
11.2.3 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.3 Model forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
11.3.1 Distribution models for tree size . . . . . . . . . . . . . . . . . . . . 342
11.3.2 Model comparison criteria . . . . . . . . . . . . . . . . . . . . . . . . 346
11.4 Mathematics of size distributions . . . . . . . . . . . . . . . . . . . . . . . . 348
11.4.1 Transformation of tree size . . . . . . . . . . . . . . . . . . . . . . . 348
11.4.2 Basal-area weighted distribution . . . . . . . . . . . . . . . . . . . . 353
11.4.3 Weighted distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 356
11.4.4 Scaling the size distribution . . . . . . . . . . . . . . . . . . . . . . . 360
11.4.5 Utilizing the arithmetic relationships of stand variables . . . . . . . 365
11.5 Modeling tree size in a population of stands . . . . . . . . . . . . . . . . . 368
11.5.1 The statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
11.5.2 PPM and PRM approaches . . . . . . . . . . . . . . . . . . . . . . . 370
11.5.3 Improving the prediction using sample information . . . . . . . . . . 373